Get the transformed value from given frame to initial frame
Slapped together a quick test script from Panel template. A custom property '["frame"]' on the object is used to hold the other frame, calculated against movement from current frame.

Script simply queries the location fcurves of the object, and uses fcurve.evaluate(frame) to retrieve the value at obj["frame"].



import bpy

class HelloWorldPanel(bpy.types.Panel):
    """Creates a Panel in the Object properties window"""
    bl_label = "Hello World Panel"
    bl_idname = "OBJECT_PT_hello"
    bl_space_type = 'PROPERTIES'
    bl_region_type = 'WINDOW'
    bl_context = "object"

    @classmethod
    def poll(cls, context):
        obj = context.object
        if "frame" not in obj.keys():
            return False
        if not hasattr(obj, "animation_data"):
            return False
        if not obj.animation_data.action:
            return False
        return True

    def draw(self, context):
        obj = context.object
        scene = context.scene
        layout = self.layout
        # get current frame location
        loc_current = obj.location.copy()
        row = layout.row()
        col = row.column()
        col.prop(scene, "frame_current")
        col.prop(obj, "location")
        loc = loc_current.copy()

        # using fcurves
        frame = obj["frame"]
        action = obj.animation_data.action
        col = layout.column()

        col.label("Moved")
        col.prop(obj, '["frame"]', text="From Frame")
        row = layout.row()
        for index in [0, 1, 2]:
            fcurve = action.fcurves.find('location', index)
            if fcurve:
                loc[index] = fcurve.evaluate(frame)
            row.label("%7.4f" % (loc - loc_current)[index])

def register():
    bpy.utils.register_class(HelloWorldPanel)

def unregister():
    bpy.utils.unregister_class(HelloWorldPanel)

if __name__ == "__main__":
    bpy.context.object["frame"] = 33 # set another frame in custom prop
    register()